If the sequence \(a_n\) is defined as \(a_n=n^2+n+\sqrt{n+3},\) then which of the following values of \(n\) represents the first term such that \(a_n > 500?\)
A. 13
B. 22
C. 33
D. 46
E. 78
[spoiler]OA=B[/spoiler]
Source: Magoosh
If the sequence \(a_n\) is defined as \(a_n=n^2+n+\sqrt{n+3},\) then which of the following values of \(n\) represents
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- Jay@ManhattanReview
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So we have \(n^2+n+\sqrt{n+3} > 500\)
We have to find out the minimum positive integer value of n such that \(\sqrt{n+3}\) is an integer, and \(n^2+n+\sqrt{n+3} > 500\). By hit and trial, we find that n = 22 serves the purpose as at n = 13, though \(\sqrt{n+3}\) is an integer, \(n^2+n+\sqrt{n+3} < 500\).
Correct answer: B
Hope this helps!
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Solution:
The best way to solve this problem is to try the answer choices out. Since the terms n and √(n + 3) are much smaller than n^2, we can assume a(n) = n^2.
A. If n = 13, a(13) = 13^2 = 169. This is a lot less than 500, so A can’t be the correct answer.
B. If n = 22, a(22) = 22^2 = 484. This is about 500, so let’s use the actual formula for a(n):
a(22) = 22^2 + 22 + √(22 + 3) = 484 + 22 + 5 = 511
This is greater than 500, so B is the correct answer.
Answer: B
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